Question: Solve the equation. $\dfrac{dy}{dx}=5y^2\cos(x)$ Choose 1 answer: Choose 1 answer: (Choice A) A $y=-\dfrac{5}{\sin(x)+C}$ (Choice B) B $y=-\dfrac{5}{\sin(x)}+C$ (Choice C) C $y=\dfrac{1}{-5\sin(x)+C}$ (Choice D) D $y=\dfrac{1}{-5\sin(x)}+C$
Explanation: We can bring this equation to the form $f(y)\,dy=g(x)\,dx$ : $\begin{aligned} \dfrac{dy}{dx}&=5y^2\cos(x) \\\\ \dfrac{1}{y^2}\,dy&=5\cos(x)\,dx \end{aligned}$ This means we can solve this equation using separation of variables! $\begin{aligned} \dfrac{1}{y^2}\,dy&=5\cos(x)\,dx \\\\ \int \dfrac{1}{y^2}\,dy&=\int 5\cos(x)\,dx \\\\ -\dfrac{1}{y}&=5\sin(x)+C_1 \\\\ \dfrac{1}{y}&=-5\sin(x)+C \\\\ y&=\dfrac{1}{-5\sin(x)+C} \end{aligned}$ [Where did we get C?] Notice that after the integration, more work was required in order to isolate $y$. In conclusion, this is the solution of the equation: $y=\dfrac{1}{-5\sin(x)+C}$